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Theorem xrmaxleim 11271
Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
Assertion
Ref Expression
xrmaxleim  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )

Proof of Theorem xrmaxleim
Dummy variables  f  g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrlttri3 9816 . . . 4  |-  ( ( f  e.  RR*  /\  g  e.  RR* )  ->  (
f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
21adantl 277 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  (
f  e.  RR*  /\  g  e.  RR* ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 simplr 528 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  B  e.  RR* )
4 prid2g 3712 . . . 4  |-  ( B  e.  RR*  ->  B  e. 
{ A ,  B } )
53, 4syl 14 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  B  e.  { A ,  B }
)
6 simpllr 534 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  <_  B )
7 xrlenlt 8041 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
87ad3antrrr 492 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( A  <_  B  <->  -.  B  <  A ) )
96, 8mpbid 147 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  A )
10 breq2 4022 . . . . . . 7  |-  ( y  =  A  ->  ( B  <  y  <->  B  <  A ) )
1110notbid 668 . . . . . 6  |-  ( y  =  A  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1211adantl 277 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
139, 12mpbird 167 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  y )
14 xrltnr 9798 . . . . . 6  |-  ( B  e.  RR*  ->  -.  B  <  B )
1514ad4antlr 495 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  B )
16 breq2 4022 . . . . . . 7  |-  ( y  =  B  ->  ( B  <  y  <->  B  <  B ) )
1716notbid 668 . . . . . 6  |-  ( y  =  B  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1817adantl 277 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1915, 18mpbird 167 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  y )
20 elpri 3630 . . . . 5  |-  ( y  e.  { A ,  B }  ->  ( y  =  A  \/  y  =  B ) )
2120adantl 277 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  (
y  =  A  \/  y  =  B )
)
2213, 19, 21mpjaodan 799 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  -.  B  <  y )
232, 3, 5, 22supmaxti 7022 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B )
2423ex 115 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   {cpr 3608   class class class wbr 4018   supcsup 7000   RR*cxr 8010    < clt 8011    <_ cle 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-pre-ltirr 7942  ax-pre-apti 7945
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-cnv 4649  df-iota 5193  df-riota 5847  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017
This theorem is referenced by:  xrmaxltsup  11285  xrmaxadd  11288  xrmineqinf  11296
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